A lithographic apparatus is a machine that applies a desired pattern onto a substrate, usually onto a target portion of the substrate. A lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). In that instance, a patterning device, which is alternatively referred to as a mask or a reticle, may be used to generate a circuit pattern to be formed on an individual layer of the IC. This pattern can be transferred onto a target portion (e.g., comprising part of, one, or several dies) on a substrate (e.g., a silicon wafer). Transfer of the pattern is typically via imaging onto a layer of radiation-sensitive material (resist) provided on the substrate. In general, a single substrate will contain a network of adjacent target portions that are successively patterned. Known lithographic apparatus include so-called steppers, in which each target portion is irradiated by exposing an entire pattern onto the target portion at one time, and so-called scanners, in which each target portion is irradiated by scanning the pattern through a radiation beam in a given direction (the “scanning”-direction) while synchronously scanning the substrate parallel or anti-parallel to this direction. It is also possible to transfer the pattern from the patterning device to the substrate by imprinting the pattern onto the substrate.
In order to monitor the lithographic process, it is necessary to measure parameters of the patterned substrate, for example the overlay error between successive layers formed in or on it. There are various techniques for making measurements of the microscopic structures formed in lithographic processes, including the use of scanning electron microscopes and various specialized tools. One form of specialized inspection tool is a scatterometer in which a beam of radiation is directed onto a target on the surface of the substrate and properties of the scattered or reflected beam are measured. By comparing the properties of the beam before and after it has been reflected or scattered by the substrate, the properties of the substrate can be determined. This can be done, for example, by comparing the reflected beam with data stored in a library of known measurements associated with known substrate properties. Two main types of scatterometer are known. Spectroscopic scatterometers direct a broadband radiation beam onto the substrate and measure the spectrum (intensity as a function of wavelength) of the radiation scattered into a particular narrow angular range. Angularly resolved scatterometers use a monochromatic radiation beam and measure the intensity of the scattered radiation as a function of angle.
More generally, it would be useful to be able to compare the scattered radiation with scattering behaviors predicted mathematically from models of structures, which can be freely set up and varied until the predicted behavior matches the observed scattering from a real sample. Unfortunately, although it is in principle known how to model the scattering by numerical procedures, the computational burden of the known techniques renders such techniques impractical, particularly if real-time reconstruction is desired, and/or where the structures involved are more complicated than a simple structure periodic in one-dimension.
CD reconstruction belongs to a group of problems known under the general name of inverse scattering, in which observed data is matched to a possible physical situation. The aim is to find a physical situation that gives rise to the observed data as closely as possible. In the case of scatterometry, the electromagnetic theory (Maxwell's equations) allows one to predict what will be the measured (scattered) data for a given physical situation. This is called the forward scattering problem. The inverse scattering problem is now to find the proper physical situation that corresponds to the actual measured data, which is typically a highly nonlinear problem. To solve this inverse scattering problem, a nonlinear solver is used that uses the solutions of many forward scattering problems. In known approaches for reconstruction, the nonlinear problem is founded on three ingredients:
Gauss-Newton minimization of the difference between measured data and data computed from the estimated scattering setup;
parameterized shapes in the scattering setup, e.g. radius and height of a contact hole;
sufficiently high accuracy in the solution of the forward problem (e.g. computed reflection coefficients) each time the parameters are updated.
For CD reconstruction of 1D- or 2D-periodic structures (e.g. gratings) a Volume Integral Method (VIM) can be used to efficiently compute the solution of the pertaining scattering problem, as has been disclosed in US patent application publication no. US2011/0218789 A1 and US patent application publication no. US2011/0098992 A1, which are incorporated herein by reference. The particular choice for a spectral expansion in the plane of periodicity, results in a fully decoupled modal description in the orthogonal aperiodic direction. This modal description, which takes the form of a sequence of 1-dimensional integral equations, each with a 1D Green's function that describes the wave propagation per mode, is discretized or sampled to arrive at a numerical scheme. For a low-order discretization scheme, the structure of the 1D integral equations leads to a numerically stable and efficient O(N) algorithm for the matrix-vector product, where N is the number of samples per 1D integral equation.
The Chebyshev expansion has been known for a long time and especially its relation to the discrete cosine transformation (DCT) has been exploited to produce efficient numerical schemes for a range of numerical methods including differential equations and 1D integral equations, for example [2,4,5].
A distinct disadvantage of a low-order expansion to discretize the 1D integral equations is that the numerical discretization error converges as O(h2), where h is the mesh size. For structures that are high with respect to the wavelength along the aperiodic direction, many samples are needed to reach an acceptable level of accuracy, which in turn leads to long computation times and large memory requirements.
An approach to increase the convergence rate is then to introduce higher-order expansions with high-order polynomials, which gives rise to pseudo-spectral methods, which include Chebyshev polynomial expansions and Legendre polynomial expansions. Pseudo-spectral methods are known for their exponential convergence. However, there are two caveats with this approach: Does the high-order expansion give rise to an efficient matrix-vector product?; and is the resulting numerical scheme stable?
Chebyshev expansions yield an efficient matrix-vector product of O(N log N), owing to the use of a discrete cosine transformation (DCT). However, the stability turns out to be a substantial problem, especially for evanescent modes, e.g. the scheme presented in [2] has been found to be highly unstable for the above mentioned integral equation, as illustrated by FIG. 16 herein. Without further measures for stability, the resulting linear system of integral equations has a very high condition number and therefore it cannot be reliably solved iteratively.